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Low-Temperature Phase Transition and Structure of Ordered Phase in K3H(SO4)2 (TKHS)-Family Materials
Low-Temperature Phase Transition and Structure of Ordered Phase in K3H(SO4)2 (TKHS)-Family Materials S. P. Dolin, A. A. Levin, T. Yu. Mikhailova and M. V. Solin NS Kurnakov Institute of General and Inorganic Chemistry, Russian Academy of Sciences, Leninskii pr. 31, 119991 Moscow, Russia
Abstract Suitability of quantum chemistry to describe the structural phase transitions on the semiquantitative level in H-bonded ferroelectrics and related materials has been demonstrated. The low-temperature H/D-ordering transitions have been treated for the TKHS-like materials in the scope of the static and dynamic Ising models. Parameters of the Ising-type Hamiltonians are evaluated on the Hartree–Fock/second-order perturbation theory level using the available diffraction data. The type of D-ordering in the low-temperature phase of M3D(AO4)2 specimens, which is debated so far, is proposed in the limits of the static model. As shown this ordering points out the antiferroelectric-type structure with the b-doubling of paraelectric A2=a cell and admits the c-doubling of this cell. Estimations of Tc in the fully deuterated specimens are obtained by the Bethe cluster approximation using the dynamic model. An absence of such transition for M3H(AO4)2 has been explained by both the proton quantum fluctuations (tunneling effect) and the differences in geometries of H- and D-bonds (geometric isotope effect). Contents 1. Introduction 2. Model Hamiltonian 3. Parameters V and Jij 4. Ground state of localized deuterons 5. Thermodynamics of the TKHS family Acknowledgements References
579 580 581 582 583 585 586
1. INTRODUCTION Theoretical treatment of order– disorder structural phase transitions in hydrogenbonded ferroelectrics implies, in principle, the determination of the transition critical temperature Tc and accompanying crystal structure transformations. ADVANCES IN QUANTUM CHEMISTRY, VOLUME 44 ISSN: 0065-3276 DOI 10.1016/S0065-3276(03)44038-0
q 2003 Elsevier Inc. All rights reserved
S. P. Dolin et al.
580
These important, but not completely understood, problems are considered here by using the novel, quantum chemical, approach to the microscopical theory of ferroelectrics and related materials [1]. The isomorphous H-bonded crystals M3(H/D)(AO4)2 (M ¼ K, Rb; A ¼ S, Se) are taken as examples. There are two reasons of such choice. This family is investigated actively at present. Moreover, it is a suitable subject of theoretical examination because of simple chemical constitution of the TKHS-like compounds (zero-dimensional H-bond network). The present work continues our theoretical investigations [1] of this TKHS family. In contrast to [1] the Ising model is employed here to examine the possible types of deuteron ordering in the low-temperature phase (below < 100 K) as well as to study the thermodynamic behavior of these materials on semi-quantitative level in the limits of the Bethe cluster approximation. The Jahn – Teller theory terminology is not applied explicitly in this paper. However, the strong low-barrier H-bonds in considered systems demonstrate some of the inherent properties of the typical pseudo-Jahn– Teller centers.
2. MODEL HAMILTONIAN The Ising-type Hamiltonian is usually employed for H-bonded ferroelectrics. In the more general quantum –mechanical approach (dynamic Ising model [2,3]) it has a form: H ¼ 2V
N X
s xi
i¼1
X 1 N 2 J s zs z 2 i; j¼1 ij i j
ð1Þ
Here s xi ; and s zi are the Pauli matrices, that act on the proton (deuteron) wavefunctions cL;i and cR;i localized in the left and right potential wells of the proton (deuteron) potential energy profile of the ith H-bond (in accordance with the available diffraction data the short H-bonds O – H· · ·O with two off-center sites of a proton are considered). The parameters of Hamiltonian (1), i.e., the tunneling parameter V and the Ising parameters Jij ; describe the motion of a proton (deuteron) along H-bond and the effective pair interactions of these particles, respectively. When V is substantially smaller than Jij the static approximation ðV ¼ 0Þ becomes valid. Instead of equation (1) one has in this case E¼2
X 1 N J s zs z 2 i; j¼1 ij i j
ð2Þ
In equation (2) E is the (potential) energy of the proton (deuteron) subsystem and s zi are considered as classical dihotomic variables s zi ¼ ^1: These values correspond to particles localized in one of two possible equilibrium positions on each H-bond.
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3. PARAMETERS V AND Jij The parameters of Hamiltonians (1) and (2) are determined in our approach by pure theoretical way using different quantum chemical models and calculations unlike the traditional fitting the experimental thermodynamic and dielectric data. Our method of the many-pseudospin clusters [1,4] seems to be the most reliable way of Jij determination. The latter are obtained in this case within the static approximation from the system of equations for a typical crystal fragment (cluster) for all possible proton distributions on H-bonds. The left-hand side of any equation expresses the cluster total energy in terms of Jij , while the right-hand side is determined by means of the quantum chemical calculation of this energy. Table 1 presents the J12 ; J13 ; J14 values (describing the couplings of a given pseudospin with the first, second, and third nearest neighboring pseudospins, respectively) calculated by such manner using the 8-spin cluster {1; 2; 3; 4; 10 ; 20 ; 30 ; 40 } in the TKHS-like crystals (Fig. 1). The calculation of the total energy of this cluster was performed ab initio, using the pseudopotential approximation for inner electronic shells. These calculations were carried out by the use of more accurate diffraction data (see Ref. [5] for details). The signs of Jij correspond to the choice of the positive directions ofPpseudospins depicted in Fig. 1. The values of the molecular field parameter J0 ¼ j Jij are also shown in Table 1; here J0 ¼ 2J12 þ 2J14 þ 4J13 : These values of the major Jij parameters are rather close to those obtained using the 2- and 4-spin clusters [4,5] and trend towards simultaneous decrease in each parameter on several percents with the increase in the cluster size so that the inequality J12 . lJ14 l $ J13 is satisfied in any case. The most reliable values of the calculated tunneling parameters VH and VD are presented in Table 2. These values were obtained with the help of the calculated ab initio potential energy profiles, corrected in according with the diffraction data for proton (deuteron) positions [5].
Table 1. Averaged values of the major Ising J12 ; J14 ; J13 ; and molecular field J0 parameters (K) for M3(H/D) (AO4)2, M ¼ K, Rb; A ¼ S, Se M3H(AO4)2 Specimen/model 4-Spin cluster 8-Spin cluster
a
M3D(AO4)2
J12
J14
J13
J0
J12
J14
J13
J0
85 81
231 229
24 21
204 188
140 129
2 51 2 47
39 34
334 300
The calculations were carried out by the pseudopotential method with SBK basis set (GAMESS codes [19,20]); see Refs. [4,5] for the details of calculations. a The data for the 4-spin cluster [5] are shown for comparison.
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Fig. 1. H-bonds in the crystal cell of TKHS-like crystal; the sign (þ) relates to covalently bonded oxygen and (2) to hydrogen-bonded one. The positive signs of all pseudospins chosen to correspond the dipole directions of H-bonds.
4. GROUND STATE OF LOCALIZED DEUTERONS While the crystalline structures of the TKHS-like materials were studied by diffraction methods for two decades, some important structural aspects are open to question. The use of the static Ising model along with the data of Table 1 permits to make an attempt to provide insight into urgent problem of ferroelectrics crystal chemistry concerning the actual deuteron arrangement in the low-temperature phase of M3D(AO4)2 compounds. The crystal structure of the paraelectric phase of the TKHS-like materials is commonly described by space group A2=a [6] (although there appears an evidence speaking for space group A2 [7,8]) and the corresponding H-bond positions in crystal cell are shown in Fig. 1. It is seen that this zerodimensional H-bond network can be considered as consisting of separate layers parallel to ab plane. The internal geometry of all layers is identical but they can be divided into two types – L0 and L1/2 in respect of their position in lattice. As from Fig. 1: (1) the layers of different types are alternated, (2) the nearest-neighboring layers L0 (as well as L1/2) are matched by the ^ c-translations, and (3) each L0 layer is matched with its two L1/2 neighbors by the simultaneous shifts by b=2 and ^c=2:
Table 2. Averaged values of VH=D for M3(H/D) (AO4)2
VH/D, K
M3H(AO4)2
M3H(AO4)2
Uncorrected Correction 1 Correction 2
570 ^ 70 430 ^ 50 420 ^ 80
180 ^ 40 120 ^ 20 90 ^ 15
See Ref. [5] for the details of calculations.
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In the scope of the static Ising model the pseudospin (deuteron) distribution corresponding to the lowest energy of localized deuteron system (ground state) can be found for any layer. To do it in practice, our analysis has been restricted by comparison of rather simple periodic pseudospin distributions because the correct mathematical procedure is unknown in the case of arbitrary pseudospin distributons, in particular, involving nonperiodic ones. When the most strong pseudospin interactions are taken account of (describing by J12, J14, J13; Table 1) then the ground state in point corresponds to the pseudospin configuration L0(þ ) which is schematically shown on ab plane (Fig. 1). The same energy has the L0(2 ) configuration where all pseudospins are reversed simultaneously (as well as the similar configurations L1/2(þ ) and L1/2(2 )). The estimations of coupling parameters Jij for neighboring layer pseudospins exhibit a rather weak interconnection of their pseudospin structures, so that the energy of whole lattice can be approximately presented as the sum of separate layer energies. The ground state in this case can be ascribed by the following scheme: · · ·L1=2 ð^ÞL0 ð^ÞL1=2 ð^ÞL0 ð^Þ· · ·
ð3Þ
where the signs (þ ) and (2 ) do not correlated in different layers. All of structures (3) relate to antiferroelectric phase that is in an agreement with the available measurements evidencing just the same character of low-temperature phase in the M3D(AO4)2 crystals [6]. It is also evident that the doubling of b-parameter of the paraelectric A2=a cell under transition to the low-temperature D-ordered phase directly follows from proposed scheme (3). Recently, such b-doubling is observed experimentally in [7]. This scheme is also consistent with the doubling of c-parameter of A2=a cell found in the cited paper. In particular, such b-doubling takes place when the layer sequence (3) takes the form: {L0 ðþÞL1=2 ð^ÞL0 ð2ÞL1=2 ð^ÞL0 ðþÞ}1
or ð4Þ
{L0 ð2ÞL1=2 ð^ÞL0 ðþÞL1=2 ð^ÞL0 ð2Þ}1 It should be noted, however, that we do not discuss many different structural changes in crystals accompanying the low-temperature D-ordering transition (displacements of metal cations, rotations of the AO4-terahedra, etc.). Therefore, our analysis is limited by specific changes in H-bond network only and does not claim on the exhaustive treatment of all crystallographic problems.
5. THERMODYNAMICS OF THE TKHS FAMILY Our quantum chemical approach has been applied to explain the thermodynamics of the TKHS-like materials on the level of simple mean (molecular) field approximation [1,4] which, however, is not well adapted to provide for quantitative estimations of the critical temperature of phase transitions. To examine the
S. P. Dolin et al.
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thermodynamic behavior of the crystals at hand at low temperatures in the range of 100 K the (Bethe) clusters approximations [3,9,10] has been applied. The latter is a natural generalization of the molecular field approximation because it allows us to have regard for proton– proton correlation effect, at least in part, and therefore it is believed to be more reliable for the quantitative treatment. In Refs. [9,10], the cluster approximation has been developed in the scope of the dynamic model (1) using KH2PO4 (KDP) as an example. Here the numerical version [10] of this approximation is employed. The four-particle Hamiltonian for the {1; 2; 3; 4} cluster H4 ¼ 2V
4 X
s xi 2
i¼1
X 4 X 1 4 Jij s zi s zj 2 nw s zi 2 i;j¼1 i¼1
as well as the single-particle Hamiltonian H1 ¼ 2Vs x1 2 ws z1
ð5Þ
ð6Þ
are considered with this end in view. Here w is the (variational) parameter of the mean field acting on the pseudospins 1– 4 in equation (5) and on the pseudospin 1 in equation (6) from remaining pseudospins of crystal. This mean field in Hamiltonian (5) is reduced by the factor
n¼12
J12 þ J14 þ J13 < 5=8 J0
because the interactions of a given pseudospin with three nearest-neighboring pseudospins inside the cluster have already taken into account in the second term of equation (5). Then, the use of the coincidence condition for the thermal expectation value(s) ks zi l obtained with Hamiltonians (5) and (6) results in the following consistency equation qffiffiffiffiffiffiffiffiffiffiffi V2 þ w2 w f ks zi l ¼ qffiffiffiffiffiffiffiffiffiffiffi th ¼ 16 ð7Þ X kB T V2 þ w2 expð2li =kB TÞ i¼1
Here
* + 16 4 X 1X z f ¼ Fi s j Fi expð2li =kB TÞ 4 i¼1 j¼1
and
Fi ¼
16 X
Cji Qj
j¼1
where Qi and Fi are the basis sets, that diagonalize the matrix of the operator (5) in the cases V ¼ 0 and V – 0; respectively, and li are the eigenvalues of Hamiltonian (5) at V – 0:
Phase Transition in the K3H(SO4)2 Family
585
Table 3. Estimations of Tc for M3(H/D) (AO4)2 Tc , K Uncorrected Correction 1 Correction 2 Experiment
M3H(AO4)2
M3D(AO4)2
– – – (20a)
– # 115 95 4 135 70 4 115
Jij obtained from 8-spin cluster (Table 1) and VD (Table 2) are used. a According to Ref. [13].
The critical temperature Tc of the transition from paraelectric phase to ordered phase is obtained by means of the numerical solution of (7). Table 3 gathers the Tc -values calculated for the TKHS-like crystals with the help of the data from Tables 1 and 2. One can see the reasonable agreement with the available experimental data [11,12] for the deuterated compounds. Meanwhile, equation (7) does not lead to some sort Tc for the M3H(AO4)2 specimens, that agrees with the tendency to an absence of the low-temperature H-ordering phase transition for such materials up to helium temperatures [6,13]. The reason of such a noticeable isotope effect may be easily clarified. It can be seen that the Tc estimate goes to zero with the growth of V for given Jij values. On the contrary, the Tc value grows together with Jij ; for a given V: Hence, an absence of the low-temperature transition to the proton-ordered phase can be explained by the quantum paraelectric behavior of these systems, i.e., the proton ordering suppression due to proton quantum fluctuations in lattice. There are two alternative viewpoints to account for the large isotope effect on the critical temperature in H-bonded ferroelectrics. The first is the H/D tunneling effect [14,15] and the second is the ‘geometric isotope effect’ [16 –18], which is based on a correlation between Tc and the differences in geometry of O – H· · ·O and O –D· · ·O bonds. Our treatment of the TKHS family, in which these structural changes are explicitly incorporated at the very beginning, demonstrates that both tunneling and H-bond geometry play an essential role in the description of the origin of the H/D-ordering transitions, in particular, the quantum paraelecticity nature. It should be marked in conclusion that the explanation of some delicate peculiarities of the TKHS-like materials (even on the semi-quantitative level), such as the x – T phase diagrams for the mixed M3HxD12x(AO4)2 compounds, possible transition in K3H(SeO4)2, etc., requires, of course, more detail examinations, involving at least some of proton isotopes –lattice coupling. ACKNOWLEDGEMENTS The authors are indebted to Professors: Z. Smedarchina, A. I. Baranov, F. Fillaux, K. Gesi, and V. G. Vaks for the fruitful discussions, devoted different aspects of our study. This work is supported by the Russian Foundation for Basic Research, Project 02-03-32160.
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REFERENCES [1] S. P Dolin, A. A Levin, T. Yu Mikhailova and M. V. Solin, in Vibronic Interaction: Jahn–Teller Effect in Crystals and Molecules (eds M. D. Kaplan and G. O. Zimmerman), Kluwer Academic Publishers, Dordrecht, 2000, p. 263. [2] M. E. Lines and A. M. Glass, Glass, Principles and Application of Ferroelectrics and Related Materials, Clarendon Press, Oxford, 1977. [3] V. G. Vaks, Introduction to Microscopical Theory of Ferroelectrics, Nauka, Moscow, 1973, in Russian. [4] S. P. Dolin, A. A. Levin, T. Yu. Mikhailova, M. V. Solin and L. I. Trakhtenberg, Int. J. Quantum Chem., 2002, 88 (4), 463. [5] S. P. Dolin, A. A. Levin, T. Yu. Mikhailova, M. V. Solin and N. S. Strokach, Int. J. Quantum Chem., in press. [6] M. Ichikawa, T. Gustafsson and I. Olovsson, Ferroelectrics, 1994, 159, 73. [7] I. Tamura and Y. Noda, Ferroelectrics, 1998, 219, 135. [8] T. Gustafsson, M. Ichilawa and I. Olovsson, Solid State Commun., 2000, 115, 473. [9] V. G. Vaks and V. I. Zinenko, JETP, 1973, 64, 650, in Russian. [10] R. Blinc and S. Svetina, Phys. Rev., 1966, 147 (2), 430. [11] T. Matsuo, A. Inaba, O. Yamamuro and N. Onoda-Yamamuro, J. Phys. Condens. Matter., 2000, 12, 8595. [12] N. Onoda-Yamamuro, O. Yamamura, T. Matsuo, M. Ichikawa, R. M. Ibberson and W. I. F. David, J. Phys. Condens. Matter., 2000, 12, 8559. [13] M. Endo, T. Kaneko, Y. Osaka and Y. Makita, J. Phys. Soc. Jpn, 1983, 52, 3829. [14] R. Blinc, J. Phys. Chem. Solids, 1960, 13, 204. [15] R. Blinc and B. Z˘eks˘, Soft Modes in Ferroelectrics and Antiferroelectrics, North-Holland, Amsterdam, 1974. [16] M. Ichikawa, Chem. Phys. Lett., 1971, 27, 103. [17] M. Ichikawa, Phys. Rev., 1981, B36, 874. [18] M. Ichikawa, Ferroelectrics, 1995, 168, 177. [19] M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis and J. A. Montgomery, J. Comput. Chem., 1993, 14, 1347. [20] A. A. Granovsky, URL http://clasiic.chem.msu.su/gran/gamess/index.html
Abstract Suitability of quantum chemistry to describe the structural phase transitions on the semiquantitative level in H-bonded ferroelectrics and related materials has been demonstrated. The low-temperature H/D-ordering transitions have been treated for the TKHS-like materials in the scope of the static and dynamic Ising models. Parameters of the Ising-type Hamiltonians are evaluated on the Hartree–Fock/second-order perturbation theory level using the available diffraction data. The type of D-ordering in the low-temperature phase of M3D(AO4)2 specimens, which is debated so far, is proposed in the limits of the static model. As shown this ordering points out the antiferroelectric-type structure with the b-doubling of paraelectric A2=a cell and admits the c-doubling of this cell. Estimations of Tc in the fully deuterated specimens are obtained by the Bethe cluster approximation using the dynamic model. An absence of such transition for M3H(AO4)2 has been explained by both the proton quantum fluctuations (tunneling effect) and the differences in geometries of H- and D-bonds (geometric isotope effect). Contents 1. Introduction 2. Model Hamiltonian 3. Parameters V and Jij 4. Ground state of localized deuterons 5. Thermodynamics of the TKHS family Acknowledgements References
579 580 581 582 583 585 586
1. INTRODUCTION Theoretical treatment of order– disorder structural phase transitions in hydrogenbonded ferroelectrics implies, in principle, the determination of the transition critical temperature Tc and accompanying crystal structure transformations. ADVANCES IN QUANTUM CHEMISTRY, VOLUME 44 ISSN: 0065-3276 DOI 10.1016/S0065-3276(03)44038-0
q 2003 Elsevier Inc. All rights reserved
S. P. Dolin et al.
580
These important, but not completely understood, problems are considered here by using the novel, quantum chemical, approach to the microscopical theory of ferroelectrics and related materials [1]. The isomorphous H-bonded crystals M3(H/D)(AO4)2 (M ¼ K, Rb; A ¼ S, Se) are taken as examples. There are two reasons of such choice. This family is investigated actively at present. Moreover, it is a suitable subject of theoretical examination because of simple chemical constitution of the TKHS-like compounds (zero-dimensional H-bond network). The present work continues our theoretical investigations [1] of this TKHS family. In contrast to [1] the Ising model is employed here to examine the possible types of deuteron ordering in the low-temperature phase (below < 100 K) as well as to study the thermodynamic behavior of these materials on semi-quantitative level in the limits of the Bethe cluster approximation. The Jahn – Teller theory terminology is not applied explicitly in this paper. However, the strong low-barrier H-bonds in considered systems demonstrate some of the inherent properties of the typical pseudo-Jahn– Teller centers.
2. MODEL HAMILTONIAN The Ising-type Hamiltonian is usually employed for H-bonded ferroelectrics. In the more general quantum –mechanical approach (dynamic Ising model [2,3]) it has a form: H ¼ 2V
N X
s xi
i¼1
X 1 N 2 J s zs z 2 i; j¼1 ij i j
ð1Þ
Here s xi ; and s zi are the Pauli matrices, that act on the proton (deuteron) wavefunctions cL;i and cR;i localized in the left and right potential wells of the proton (deuteron) potential energy profile of the ith H-bond (in accordance with the available diffraction data the short H-bonds O – H· · ·O with two off-center sites of a proton are considered). The parameters of Hamiltonian (1), i.e., the tunneling parameter V and the Ising parameters Jij ; describe the motion of a proton (deuteron) along H-bond and the effective pair interactions of these particles, respectively. When V is substantially smaller than Jij the static approximation ðV ¼ 0Þ becomes valid. Instead of equation (1) one has in this case E¼2
X 1 N J s zs z 2 i; j¼1 ij i j
ð2Þ
In equation (2) E is the (potential) energy of the proton (deuteron) subsystem and s zi are considered as classical dihotomic variables s zi ¼ ^1: These values correspond to particles localized in one of two possible equilibrium positions on each H-bond.
Phase Transition in the K3H(SO4)2 Family
581
3. PARAMETERS V AND Jij The parameters of Hamiltonians (1) and (2) are determined in our approach by pure theoretical way using different quantum chemical models and calculations unlike the traditional fitting the experimental thermodynamic and dielectric data. Our method of the many-pseudospin clusters [1,4] seems to be the most reliable way of Jij determination. The latter are obtained in this case within the static approximation from the system of equations for a typical crystal fragment (cluster) for all possible proton distributions on H-bonds. The left-hand side of any equation expresses the cluster total energy in terms of Jij , while the right-hand side is determined by means of the quantum chemical calculation of this energy. Table 1 presents the J12 ; J13 ; J14 values (describing the couplings of a given pseudospin with the first, second, and third nearest neighboring pseudospins, respectively) calculated by such manner using the 8-spin cluster {1; 2; 3; 4; 10 ; 20 ; 30 ; 40 } in the TKHS-like crystals (Fig. 1). The calculation of the total energy of this cluster was performed ab initio, using the pseudopotential approximation for inner electronic shells. These calculations were carried out by the use of more accurate diffraction data (see Ref. [5] for details). The signs of Jij correspond to the choice of the positive directions ofPpseudospins depicted in Fig. 1. The values of the molecular field parameter J0 ¼ j Jij are also shown in Table 1; here J0 ¼ 2J12 þ 2J14 þ 4J13 : These values of the major Jij parameters are rather close to those obtained using the 2- and 4-spin clusters [4,5] and trend towards simultaneous decrease in each parameter on several percents with the increase in the cluster size so that the inequality J12 . lJ14 l $ J13 is satisfied in any case. The most reliable values of the calculated tunneling parameters VH and VD are presented in Table 2. These values were obtained with the help of the calculated ab initio potential energy profiles, corrected in according with the diffraction data for proton (deuteron) positions [5].
Table 1. Averaged values of the major Ising J12 ; J14 ; J13 ; and molecular field J0 parameters (K) for M3(H/D) (AO4)2, M ¼ K, Rb; A ¼ S, Se M3H(AO4)2 Specimen/model 4-Spin cluster 8-Spin cluster
a
M3D(AO4)2
J12
J14
J13
J0
J12
J14
J13
J0
85 81
231 229
24 21
204 188
140 129
2 51 2 47
39 34
334 300
The calculations were carried out by the pseudopotential method with SBK basis set (GAMESS codes [19,20]); see Refs. [4,5] for the details of calculations. a The data for the 4-spin cluster [5] are shown for comparison.
S. P. Dolin et al.
582
Fig. 1. H-bonds in the crystal cell of TKHS-like crystal; the sign (þ) relates to covalently bonded oxygen and (2) to hydrogen-bonded one. The positive signs of all pseudospins chosen to correspond the dipole directions of H-bonds.
4. GROUND STATE OF LOCALIZED DEUTERONS While the crystalline structures of the TKHS-like materials were studied by diffraction methods for two decades, some important structural aspects are open to question. The use of the static Ising model along with the data of Table 1 permits to make an attempt to provide insight into urgent problem of ferroelectrics crystal chemistry concerning the actual deuteron arrangement in the low-temperature phase of M3D(AO4)2 compounds. The crystal structure of the paraelectric phase of the TKHS-like materials is commonly described by space group A2=a [6] (although there appears an evidence speaking for space group A2 [7,8]) and the corresponding H-bond positions in crystal cell are shown in Fig. 1. It is seen that this zerodimensional H-bond network can be considered as consisting of separate layers parallel to ab plane. The internal geometry of all layers is identical but they can be divided into two types – L0 and L1/2 in respect of their position in lattice. As from Fig. 1: (1) the layers of different types are alternated, (2) the nearest-neighboring layers L0 (as well as L1/2) are matched by the ^ c-translations, and (3) each L0 layer is matched with its two L1/2 neighbors by the simultaneous shifts by b=2 and ^c=2:
Table 2. Averaged values of VH=D for M3(H/D) (AO4)2
VH/D, K
M3H(AO4)2
M3H(AO4)2
Uncorrected Correction 1 Correction 2
570 ^ 70 430 ^ 50 420 ^ 80
180 ^ 40 120 ^ 20 90 ^ 15
See Ref. [5] for the details of calculations.
Phase Transition in the K3H(SO4)2 Family
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In the scope of the static Ising model the pseudospin (deuteron) distribution corresponding to the lowest energy of localized deuteron system (ground state) can be found for any layer. To do it in practice, our analysis has been restricted by comparison of rather simple periodic pseudospin distributions because the correct mathematical procedure is unknown in the case of arbitrary pseudospin distributons, in particular, involving nonperiodic ones. When the most strong pseudospin interactions are taken account of (describing by J12, J14, J13; Table 1) then the ground state in point corresponds to the pseudospin configuration L0(þ ) which is schematically shown on ab plane (Fig. 1). The same energy has the L0(2 ) configuration where all pseudospins are reversed simultaneously (as well as the similar configurations L1/2(þ ) and L1/2(2 )). The estimations of coupling parameters Jij for neighboring layer pseudospins exhibit a rather weak interconnection of their pseudospin structures, so that the energy of whole lattice can be approximately presented as the sum of separate layer energies. The ground state in this case can be ascribed by the following scheme: · · ·L1=2 ð^ÞL0 ð^ÞL1=2 ð^ÞL0 ð^Þ· · ·
ð3Þ
where the signs (þ ) and (2 ) do not correlated in different layers. All of structures (3) relate to antiferroelectric phase that is in an agreement with the available measurements evidencing just the same character of low-temperature phase in the M3D(AO4)2 crystals [6]. It is also evident that the doubling of b-parameter of the paraelectric A2=a cell under transition to the low-temperature D-ordered phase directly follows from proposed scheme (3). Recently, such b-doubling is observed experimentally in [7]. This scheme is also consistent with the doubling of c-parameter of A2=a cell found in the cited paper. In particular, such b-doubling takes place when the layer sequence (3) takes the form: {L0 ðþÞL1=2 ð^ÞL0 ð2ÞL1=2 ð^ÞL0 ðþÞ}1
or ð4Þ
{L0 ð2ÞL1=2 ð^ÞL0 ðþÞL1=2 ð^ÞL0 ð2Þ}1 It should be noted, however, that we do not discuss many different structural changes in crystals accompanying the low-temperature D-ordering transition (displacements of metal cations, rotations of the AO4-terahedra, etc.). Therefore, our analysis is limited by specific changes in H-bond network only and does not claim on the exhaustive treatment of all crystallographic problems.
5. THERMODYNAMICS OF THE TKHS FAMILY Our quantum chemical approach has been applied to explain the thermodynamics of the TKHS-like materials on the level of simple mean (molecular) field approximation [1,4] which, however, is not well adapted to provide for quantitative estimations of the critical temperature of phase transitions. To examine the
S. P. Dolin et al.
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thermodynamic behavior of the crystals at hand at low temperatures in the range of 100 K the (Bethe) clusters approximations [3,9,10] has been applied. The latter is a natural generalization of the molecular field approximation because it allows us to have regard for proton– proton correlation effect, at least in part, and therefore it is believed to be more reliable for the quantitative treatment. In Refs. [9,10], the cluster approximation has been developed in the scope of the dynamic model (1) using KH2PO4 (KDP) as an example. Here the numerical version [10] of this approximation is employed. The four-particle Hamiltonian for the {1; 2; 3; 4} cluster H4 ¼ 2V
4 X
s xi 2
i¼1
X 4 X 1 4 Jij s zi s zj 2 nw s zi 2 i;j¼1 i¼1
as well as the single-particle Hamiltonian H1 ¼ 2Vs x1 2 ws z1
ð5Þ
ð6Þ
are considered with this end in view. Here w is the (variational) parameter of the mean field acting on the pseudospins 1– 4 in equation (5) and on the pseudospin 1 in equation (6) from remaining pseudospins of crystal. This mean field in Hamiltonian (5) is reduced by the factor
n¼12
J12 þ J14 þ J13 < 5=8 J0
because the interactions of a given pseudospin with three nearest-neighboring pseudospins inside the cluster have already taken into account in the second term of equation (5). Then, the use of the coincidence condition for the thermal expectation value(s) ks zi l obtained with Hamiltonians (5) and (6) results in the following consistency equation qffiffiffiffiffiffiffiffiffiffiffi V2 þ w2 w f ks zi l ¼ qffiffiffiffiffiffiffiffiffiffiffi th ¼ 16 ð7Þ X kB T V2 þ w2 expð2li =kB TÞ i¼1
Here
* + 16 4 X 1X z f ¼ Fi s j Fi expð2li =kB TÞ 4 i¼1 j¼1
and
Fi ¼
16 X
Cji Qj
j¼1
where Qi and Fi are the basis sets, that diagonalize the matrix of the operator (5) in the cases V ¼ 0 and V – 0; respectively, and li are the eigenvalues of Hamiltonian (5) at V – 0:
Phase Transition in the K3H(SO4)2 Family
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Table 3. Estimations of Tc for M3(H/D) (AO4)2 Tc , K Uncorrected Correction 1 Correction 2 Experiment
M3H(AO4)2
M3D(AO4)2
– – – (20a)
– # 115 95 4 135 70 4 115
Jij obtained from 8-spin cluster (Table 1) and VD (Table 2) are used. a According to Ref. [13].
The critical temperature Tc of the transition from paraelectric phase to ordered phase is obtained by means of the numerical solution of (7). Table 3 gathers the Tc -values calculated for the TKHS-like crystals with the help of the data from Tables 1 and 2. One can see the reasonable agreement with the available experimental data [11,12] for the deuterated compounds. Meanwhile, equation (7) does not lead to some sort Tc for the M3H(AO4)2 specimens, that agrees with the tendency to an absence of the low-temperature H-ordering phase transition for such materials up to helium temperatures [6,13]. The reason of such a noticeable isotope effect may be easily clarified. It can be seen that the Tc estimate goes to zero with the growth of V for given Jij values. On the contrary, the Tc value grows together with Jij ; for a given V: Hence, an absence of the low-temperature transition to the proton-ordered phase can be explained by the quantum paraelectric behavior of these systems, i.e., the proton ordering suppression due to proton quantum fluctuations in lattice. There are two alternative viewpoints to account for the large isotope effect on the critical temperature in H-bonded ferroelectrics. The first is the H/D tunneling effect [14,15] and the second is the ‘geometric isotope effect’ [16 –18], which is based on a correlation between Tc and the differences in geometry of O – H· · ·O and O –D· · ·O bonds. Our treatment of the TKHS family, in which these structural changes are explicitly incorporated at the very beginning, demonstrates that both tunneling and H-bond geometry play an essential role in the description of the origin of the H/D-ordering transitions, in particular, the quantum paraelecticity nature. It should be marked in conclusion that the explanation of some delicate peculiarities of the TKHS-like materials (even on the semi-quantitative level), such as the x – T phase diagrams for the mixed M3HxD12x(AO4)2 compounds, possible transition in K3H(SeO4)2, etc., requires, of course, more detail examinations, involving at least some of proton isotopes –lattice coupling. ACKNOWLEDGEMENTS The authors are indebted to Professors: Z. Smedarchina, A. I. Baranov, F. Fillaux, K. Gesi, and V. G. Vaks for the fruitful discussions, devoted different aspects of our study. This work is supported by the Russian Foundation for Basic Research, Project 02-03-32160.
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